Vortices for lake equations (review with questions and speculations)

TL;DR

This paper reviews the geometric mechanics of lake equations, exploring vortex dynamics, Green functions, and conformal mappings, with implications for understanding rip currents and related phenomena.

Contribution

It provides a conceptual review of vortex behavior in lake equations, connecting geometric mechanics, Green functions, and quasiconformal mappings, and raises questions for future research.

Findings

Vortex pairs can produce rip currents near beaches.

Green functions are key to understanding stream functions in lake equations.

Connections to elasticity and conformal mappings offer new analytical perspectives.

Abstract

The `lake equation' on a planar domain D with bathymetry b(x,y) is given by $ \partial_t u + (u \cdot {\rm grad}) u= -{\rm grad}\, p \,, \,\,{\rm div} (b u) = 0 \,,\, \text{with}\,\, u \parallel \partial D.$ % \, \,\, \,\,\, \text{),}$$ We focus on Geometric Mechanics aspects, glossing over hard analysis issues. % related to the desingularization. Motivating example is a `rip current' produced by vortex pairs near a beach shore. For uniform slope beach there is a perfect analogy with \ Thomson's vortex rings. The stream function produced by a vortex is defined as the Green function of the operator $- {\rm div} ( {\rm grad} \psi/b)$ with Dirichlet boundary conditions. As in elasticity, the lake equations give rise to pseudoanalytical functions and quasiconformal mappings. Uniformly elliptic equations on closed Riemann surfaces could be called `planet equations'.